Linear algebraic groups and invariant theory

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Solving polynomial equations is one of the oldest problems in mathematics. The natural approach to this problem is to look for a sequence of substitutions (otherwise known as Tschirnhaus transformations) that simplifies this equation to a form that can be easily solved. The antient Babylonians knew how to do this for quadratic equations as early as 1600 BC. A similar approach led to the solution of cubic and quartic equations during the Renaissance. In the early 19th century Abel and Galois showed that a general polynomial equation of degree higher than four cannot be solved in radicals. One can nevertheless ask how far one can simplify this equation by Tschirnhaus transformations. Ten years ago, thinking about this question has led me and my collaborators to the notion of essential dimension, a concept that proved to be fruitful both within and far beyond the theory of polynomials. One of the goals of this proposal is to continue this research in two exciting new directions, one in the traditional setting of algebraic groups, theother in the new setting of algebraic stacks.